In many applications, we desire neural networks to exhibit invariance or equivari-ance to certain groups due to symmetries inherent in the data. Recently, frame-averaging methods emerged to be a unified framework for attaining symmetries efficiently by averaging over input-dependent subsets of the group, i.e., frames. What we currently lack is a principled understanding of the design of frames.

DeepSDF: Learning Continuous Signed Distance Functionsfor Shape Representation.

Let ψ: X Y be an arbitrary G equivariant function. We leave proving this as a future work. We now show the following: Proposition 3. The proposed distribution p We now show the following: Proposition 6. From Eq. (29), we have: ϕ Proposition 7. The proposed symmetrization From Eq. (29), we have: ϕ This is after handling the translation component of the Euclidean group E ( d) / SE (d) as in Eq. (29). We now show the following: Proposition 8. Therefore, probabilistic symmetrization can become frame averaging.

Canonicalization is a widely used strategy in equivariant machine learning, enforcing symmetry in neural networks by mapping each input to a standard form. Yet, it often introduces discontinuities that can affect stability during training, limit generalization, and complicate universal approximation theorems. In this paper, we address this by introducing adaptive canonicalization, a general framework in which the canonicalization depends both on the input and the network. Specifically, we present the adaptive canonicalization based on prior maximization, where the standard form of the input is chosen to maximize the predictive confidence of the network. We prove that this construction yields continuous and symmetry-respecting models that admit universal approximation properties. We propose two applications of our setting: (i) resolving eigenbasis ambiguities in spectral graph neural networks, and (ii) handling rotational symmetries in point clouds. We empirically validate our methods on molecular and protein classification, as well as point cloud classification tasks. Our adaptive canonicalization outperforms the three other common solutions to equivariant machine learning: data augmentation, standard canonicalization, and equivariant architectures.

Symmetry-aware methods for machine learning, such as data augmentation and equivariant architectures, encourage correct model behavior on all transformations (e.g. rotations or permutations) of the original dataset. These methods can improve generalization and sample efficiency, under the assumption that the transformed datapoints are highly probable, or "important", under the test distribution. In this work, we develop a method for critically evaluating this assumption. In particular, we propose a metric to quantify the amount of anisotropy, or symmetry-breaking, in a dataset, via a two-sample neural classifier test that distinguishes between the original dataset and its randomly augmented equivalent. We validate our metric on synthetic datasets, and then use it to uncover surprisingly high degrees of alignment in several benchmark point cloud datasets. We show theoretically that distributional symmetry-breaking can actually prevent invariant methods from performing optimally even when the underlying labels are truly invariant, as we show for invariant ridge regression in the infinite feature limit. Empirically, we find that the implication for symmetry-aware methods is dataset-dependent: equivariant methods still impart benefits on some anisotropic datasets, but not others. Overall, these findings suggest that understanding equivariance -- both when it works, and why -- may require rethinking symmetry biases in the data.

Monocular 3D pose estimators produce camera-centered skeletons, creating view-dependent kinematic signals that complicate comparative analysis in applications such as health and sports science. We present 3DPCNet, a compact, estimator-agnostic module that operates directly on 3D joint coordinates to rectify any input pose into a consistent, body-centered canonical frame. Its hybrid encoder fuses local skeletal features from a graph convolutional network with global context from a transformer via a gated cross-attention mechanism. From this representation, the model predicts a continuous 6D rotation that is mapped to an $SO(3)$ matrix to align the pose. We train the model in a self-supervised manner on the MM-Fi dataset using synthetically rotated poses, guided by a composite loss ensuring both accurate rotation and pose reconstruction. On the MM-Fi benchmark, 3DPCNet reduces the mean rotation error from over 20$^{\circ}$ to 3.4$^{\circ}$ and the Mean Per Joint Position Error from ~64 mm to 47 mm compared to a geometric baseline. Qualitative evaluations on the TotalCapture dataset further demonstrate that our method produces acceleration signals from video that show strong visual correspondence to ground-truth IMU sensor data, confirming that our module removes viewpoint variability to enable physically plausible motion analysis.

General-purpose robotic skills from end-to-end demonstrations often leads to task-specific policies that fail to generalize beyond the training distribution. Therefore, we introduce FunCanon, a framework that converts long-horizon manipulation tasks into sequences of action chunks, each defined by an actor, verb, and object. These chunks focus policy learning on the actions themselves, rather than isolated tasks, enabling compositionality and reuse. To make policies pose-aware and category-general, we perform functional object canonicalization for functional alignment and automatic manipulation trajectory transfer, mapping objects into shared functional frames using affordance cues from large vision language models. An object centric and action centric diffusion policy FuncDiffuser trained on this aligned data naturally respects object affordances and poses, simplifying learning and improving generalization ability. Experiments on simulated and real-world benchmarks demonstrate category-level generalization, cross-task behavior reuse, and robust sim2real deployment, showing that functional canonicalization provides a strong inductive bias for scalable imitation learning in complex manipulation domains. Details of the demo and supplemental material are available on our project website https://sites.google.com/view/funcanon.